Scarcity rent reconsidered

Resources

and Energy

11 (1989) 93-101.

SCARCITY

North-Holland

RENT

The Effects

RECONSIDERED of Exploration

Mine K. YUCEL Louisiana

State Unwersity,

Received

November

The scarctty of a natural resource extractive output, the product price shadow price of the resource in-situ such a case, the correct specttication the resource in-sttu and the shadow

Baton Rouge. LA 70803-6306. USA

1987, final version

received October

1988

has been measured by, among others, the unit cost of and the shadow price of the resource. We argue that the overstates the scarcity rent when exploratron is present. In of scarcity rent is a weighted average of the shadow price of prtce of future discoveries.

1. Introduction Considerable attention has been devoted to analyzing the scarcity of natural resources. The measure of scarcity is very important since it mirrors the exhaustion of the resource. Several different measures for scarcity have been advocated in the literature. The first measure is the unit cost of extractive output as used by Barnett and Morse (1963); the second is the real product price of the resource [Barnett and Morse (1963), Hertindahl (1959) Nordhaus (1974)]; and the third is the rent or shadow price of the resource [Pindyck (1978), Fisher (1979) Devarajan and Fisher (1982)]. The consensus is that the resource rent, or shadow price of the resource in-situ is the preferred measure. Using a two-period model, Devarajan and Fisher show that the shadow price of the resource in-situ can be approximated by the marginal exploration cost if exploratory output does not depend upon the stock of cumulative discoveries, When exploration is present, however, the resource rent should not be defined as merely the shadow price of the resource in-situ. The shadow price of the resource in-situ overstates the scarcity rent in such a case because the resource is no longer fixed and can be increased through exploratory effort. The resource becomes less scarce as discoveries increase. Yet, there are other factors that lead to the scarcity of the resource in cases involving exploration. Scarcity can be caused by an upper bound on discoveries or by the inverse relationship between current discoveries and the stock of cumulative dis0165-0572/89/S3.50

cl 1989, Elsevier Science Publishers

B.V. (North-Holland)

94

M.K. Wcel, Scarcity rent reconsidered

coveries. The finiteness of the resource causes exploratory output to decrease as cumulative discoveries increase. The higher the discoveries, the harder it becomes to find additional resource units. In this paper we develop a generalized measure that more closely reflects the scarcity of the resource in such a case, but reduces to the shadow price of the resource in-situ in the limiting case without exploration.

2. Model The model is a dynamic in which a competitive maximizes profits over a existing reserves of the following optimal control

subject

to the following

I; = G(x, w) -

optimization model of extraction and exploration lit-m owns reserves of a natural resource and time horizon 7: The producer, who adds to the natural resource through exploration, solves the problem:

constraints:

q( I, R),

(la)

(lb)

i = G(x, w) RZO,

x10,

120,

~20.

(14

The control variables in this problem are the level of extractive effort, I, and the level of exploratory effort, w. Production is dependent on both the level of reserves and the amount of extractive effort. The reserve level, R, is depleted through production and increased through new discoveries, G. Discoveries are dependent upon exploratory effort, w, and the level of cumulative discoveries from past discoveries, x. It is assumed that G is a decreasing returns to scale function and that 0~ G,< 1 but G,O, this would imply that increased discoveries result in higher future discoveries. This cannot be true throughout the time horizon. It could hold in the initial periods where there were positive information externalities, but not in later periods when the resource gets scarce. Putting an upper bound on total discoveries does not change the results as long as the marginal product of cumulative discoveries, G,, is negative. It is conceivable that the upper bound on x could be binding, given a relatively

M.K. Yticel, Scarcity rent reconsidered

95

long time horizon. In this case, instead of using a fixed time horizon, the final time can be made a parameter of the problem and the optimal final time can be found. These results are given in the appendix. The cost of production C, is a function of the level of extractive effort 1 and the exogenous cost of effort pr. Although extraction costs seem to be solely a function of extractive effort, they are dependent on both current and cumulative extraction and exploration. Since the reserve level is a function of cumulative production and exploration, exploration decreases the cost of production by increasing the reserve level. The cost of exploration C2 is directly proportional to the level of exploratory activity and its exogenous cost p2. The Hamiltonian associated with this maximization is

(2) The first-order

conditions

for a maximum

are (3)

H,= -he-"+G,(A,+&)=O, iI=

-H,=

&=-H,= From eq. (3), noting P-MC, From

-PqRemr’+A,q,,

(5)

-G.&l,+&).

(6)

that MC, =pJqt,

=il

we obtain

err.

eq. (4), again noting (A, +i2)ert=

(4)

(7) that MC, =pJG,,,,

MC’,.

Eq. (7) can then be written

(8) as

P-(MC,+MC,)=

-&e”.

(9)

Here 1, is the current value of the shadow price of the resource in-situ and j-Z is the current value of the shadow price of additional reserves from exploration. It is seen from eqs. (7) and (9) that the market clearing condition in this model is no longer P= MC, but can be expressed as either P= MC, +i,, err

or

P=MC,+MC,+(-&err).

96

M.K. ru’cel, Scarcity

rent reconsidered

If rent is defined as the difference between price and marginal cost, then both A1 and A2 can measure rent, depending upon the definition of cost. Marginal rents would be A1 (marginal rent 1) if only extraction costs were considered in marginal cost (as in some of the previously cited studies). However, if both the cost of extraction and the cost of exploration are included in the definition of cost [see Church (1982)], then (-A,) would be the appropriate marginal rent (marginal rent 2). We see from eqs. (5) and (7) that I, is positive and decreases with time. From eq. (9) we see that I, is always negative and always less than A1 in absolute magnitude. Eq. (6) tells us that A2 will increase with time unless G, >O. Since 1, is negative, this implies that the magnitude of 1, would be increasing toward the end of the time horizon in this case. We show in the next section that rent is a weighted average of R, and (-A,), the shadow price of the resource in-situ and the absolute magnitude of the marginal valuation of future discoveries.

3. Scarcity rent In models without exploration, the shadow price of the resource in-situ correctly specifies the scarcity rent since the resource is a fixed factor of production. But with exploration included in the model, the resource is no longer fixed, and can be increased through exploration. New reserves can now be ‘bought’ at a cost of exploration for new reserves. In this case the shadow price of the resource in-situ overstates the scarcity rent. It is easily seen from eq. (7) that if the marginal cost of production decreases as more resource is found (ceteris paribus), the shadow price of the resource would increase whereas the resource would be getting less scarce. If the shadow price of the resource in-situ is approximated by the marginal exploration cost, then this measure also would not necessarily reflect scarcity. As the resource gets scarce, exploration is curtailed and discoveries decrease. This could lead to a decrease in marginal exploration cost.’ In such a case, the measure of scarcity would become smaller as scarcity increased. The measure of scarcity rent calculated below is a weighted average of A1 and -I,,, and is a generalized measure of scarcity extending to cases with exploration. It reduces to ;1, in the special case with no exploration. The total rent, RIT: is profit minus ‘producers’ surplus’. ‘Producers’ surplus’, PS, is defined as the difference between (q x MC,) and the area under the marginal cost of production curve, and is represented by the triangular area OAB as shown in fig. 1. Without the fixed factor, profit would equal producers’ surplus and rent would equal zero. The marginal cost curve is taken to be the marginal cost of extraction. However, it can be easily shown ‘Marginal exploration costs observed in Yiicel (1986,1989).

that

decrease

toward

the end of the time horizon

have

been

M.K. Yticel, Scarcity rent reconsidered

Fig. 1. Scarcity

rent and ‘producers’

91

surplus’.

that results are not changed if the marginal cost curve is taken to be the sum of marginal extraction cost MC, and marginal discovery cost MC,. We have

RT=Pq-TC,-TC,-

qMC,-;MC,dq

.

0

Substituting

P-A,

ert for MC,

and evaluating

the integral,

RT=Pq-TC,-TC,-q(P-A,e”)+TC,,

(11)

RT=A,qe”-TC,.

(12)

We can then define per-unit p = 2, er’-

Substituting

rent as

TCJq,

for TC2 and with some manipulation

p = A1 err- wG,MC,/q.

Noting

(10)

>

(13) we obtain (14)

that MC, = (iI + &) e”, p = 2 1 err- wG,( iI + A,) elf/q,

(15)

98

M.K. Yticel, Scarcity rent reconsidered

p = A1 e”( 1 - wG,/q) + ( - A2e”)wG,/q.

(16)

Hence, the per-unit rent is a weighted average of the shadow price of reserves and the shadow price of additional finds. In other words, the per-unit rent is a weighted average of marginal rent 1 and marginal rent 2, where marginal rents are defined by eqs. (7) and (9). The weight wG,/q would be expected to be less than unity in most cases, especially toward the end of the time horizon, as the resource gets scarce. Since G is decreasing returns to scale, wG, will always be less than G. If this weight was greater than unity, it would imply that the resource found through new discoveries was greater than that extracted, in which case the reserve level R would be rising. This could happen in the initial periods if exploration was very productive, but R would have to decrease after a certain period of time. Then d ~0, and discoveries minus extractions, G-q, would be negative and hence wG,/q would be less than unity. If wG,/q was greater than unity due to high discoveries in the initial periods, the weight on 3,, would be negative, decreasing the per-unit rent. Later when d became negative high discoveries would imply that wG,/q was closer to unity and per-unit rent would be more closely approximated by (-AZ). the absolute magnitude of the shadow price of cumulative discoveries. Higher discoveries imply that the initial resource in-situ is less important and in-situ, figures less in the hence ?,, , the shadow price of the resource calculation of rent. Smaller discoveries toward the end of the time horizon reduce wG,/q. Since i., is also decreasing with time, the second term on the left-hand side decreases and per-unit rent is more closely approximated by the shadow price of the resource in-situ. After the final time period, neither future discoveries, nor the resource in-situ would be important to the profit maximizer. The producer would explore and extract until both components of rent were driven down to zero. The only case in which per-unit rent could increase would be if G, was positive. Even though extraction and exploration would wind down, with marginal extraction costs increasing and marginal exploration costs decreasing, J., would increase in absolute magnitude, thus increasing the per-unit rent toward the end of the time horizon. However, as mentioned earlier, this is not such a realistic possibility. If the model did not include exploration, the discoveries function G would be equal to zero, as would the exploration costs. We can see from eqs. (13) or (16) that in such a case, per-unit rent would exactly equal the shadow price of the resource in-situ, which is the definition of rent in earlier models. 4. Conclusion We have shown in this paper

that in resource

models

with exploration,

the

99

M.K. Yiicel, Scarcity rent reconsidered

shadow price of the resource in-situ is not the appropriate specification of scarcity rent. With exploration present, scarcity rent is a weighted average of the shadow price of the resource in-situ and the absolute magnitude of the shadow price of future discoveries. This measure of scarcity rent will reduce to the shadow price of the resource in-situ if there is no exploration. Correctly specifying the scarcity rent will prevent overstating the scarcity of a resource and will provide an improved valuation of natural resources. Appendix: The time horizon as an endogenous parameter The optimal

control

problem

with final time fixed at time t is

(1’) subject

to d=G-q,

(la’)

i=G,

(lb’)

with all variables u=

defined

as in the text. Define a new variable

u such that

t T’

Then u=O when t=O, u=l when t=T, and dt=Tdu. The problem stated in (1’) is then transformed into

(1”)

i(Pq-C,-C,)e-“Tdu, 0

subject to ;f=+(G-q),

ax

(la”)

lG

&A T

with T a parameter R=

( 1a”)

’ of the problem.

The Hamiltonian

T(Pq-p,l-p,w)e~“‘T+~l,(G-q)+~j~~G,

becomes

(2’)

M.K. Yicel, Scarcity rent reconsidered

100

The first-order

conditions

are

(3’)

(4’)

an,_ --RR=

-TPq,e-“rT++AlqR,

au

(5’)

an,_ --H,=-+G,(i,+&).

(6’)

au

Plus an additional

condition

for the parameter,

iH,du=O,

(7’)

which becomes (1 -u~T)(P~-P~I-P~w)~-“‘~-

Proceeding

in the same manner

RT=‘A

T2

’

+t,(G-q)-

+&G

1

du=O.

(8’)

as before, we obtain

eUrT-TC2

(9’)

and p

=& [A, eurT(1 -

wG,/q) +( --A2 eurT)wG,/q].

(107

Hence, the expression for rent is basically the same as in the fixed time horizon case. It can be seen from eq. (10’) that per-unit rent will increase as the parameter T increases. References Barnett, H.T. and C. Morse, 1963, Scarcity and growth: availability (Resources for the Future, Baltimore, MD). Church, Albert M., 1982, Conflicts over resource ownership

The economics (Lexington

of natural

Books,

Lexington,

resource MA).

M.K. Yicel, Scarcity rent reconsidered

101

Devarajan, S. and A.C. Fisher, 1982, Exploratton and scarcity, Journal of Political Economy 90, 127991290. Fisher, Anthony C., 1979, Measures of natural resource scarcity, m: V. Kerry Smith, ed., Scarcity and growth reconsidered (Resources for the Future, Baltimore, MD). Hertindahl, O.C., 1959, Copper costs and prices: 1870-1957 (Resources for the Future, Baltimore, MD). Nordhaus, W.D., 1974, Resources as a constramt on growth, American Economic Review 64, 22-26. Pindyck, RX, 1978, The optimal exploration and production of nonrenewable resources, Journal of Political Economy 86, 841-862. Yiicel, Mine K., 1986, Dynamic analysts of severance taxation in a competitive exhaustible resource Industry, Resources and Energy 8, no. 3, 201-218. Yiicel. Mine K.. 1989, Severance taxes and market structure m an exhaustible resource Industry, Journal of Environmental Economics and Management, forthcommg.